Mediate dominating graph of a graph

Let S be the set of minimal dominating sets of graph G and U, W ⊂ S with U ∪ W = S and U ∩ W = ∅. A Smarandachely mediate-(U, W) dominating graph [D.sub.m.sup.s] (G) of a graph G is a graph with V ([D.sub.m.sup.s] (G)) = V' = V ∪ U and two vertices u, v ∈ V' are adjacent if they are not adjacent in G or v = D is a minimal dominating set containing u. particularly, if U = S and W = ∅, i.e., a Smarandachely mediate-(S, ∅) dominating graph [D.sub.m.sup.s] (G) is called the mediate dominating graph [D.sub.m] (G) of a graph G. In this paper, some necessary and sufficient conditions are given for [D.sub.m] (G) to be connected, Eulerian, complete graph, tree and cycle respectively. It is also shown that a given graph G is a mediate dominating graph [D.sub.m] (G) of some graph. One related open problem is explored. Finally, some bounds on domination number of [D.sub.m] (G) are obtained in terms of vertices and edges of G. Key Words: Connectedness, connectivity, Eulerian, hamiltonian, dominating set, Smarandachely mediate-(U, W) dominating graph. AMS(2010): 05C69